避免线性格中的扫帚、叉子和蝴蝶
Avoiding Brooms, Forks, and Butterflies in the Linear Lattices
摘要：
设$n$是正整数，$q$是素数的幂，$\mathcal{L}_n(q)$是具有$q$元素的域上的$n$维向量空间的子空间的偏序集.这个偏序集是一个正规化的匹配偏序集，并且$\left\lfloor n/2 \right\rfloor$维的子空间集或$\left\lceil n/2 \right\rceil$维的子空间集是这个偏序集中唯一的最大反链.加强这一著名的和庆祝的结果，我们表明，除了在$\mathcal{L}_3(2)$的情况下，这些相同的子空间的集合是唯一的最大规模的家庭在$\mathcal{L}_n(q)$，避免了$\wedge$和$\vee$作为一个子集。我们将一些结果推广到扫帚和叉子，并且我们还证明了对于$k = \left\lfloor n/2 \right\rfloor$或$k = \left\lceil n/2 \right\rceil-1$，维数为$k$和$k+1$的子空间的集合的并集是$\mathcal{L}_n(q)$中唯一避免蝴蝶的最大规模族（定义见下文）。
英文摘要：
Let $n$ be a positive integer, $q$ a power of a prime, and $\mathcal{L}_n(q)$ the poset of subspaces of an $n$-dimensional vector space over a field with $q$ elements. This poset is a normalized matching poset and the set of subspaces of dimension $\left\lfloor n/2 \right\rfloor$ or those of dimension $\left\lceil n/2 \right\rceil$ are the only maximum-sized anti-chains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal{L}_3(2)$, these same collections of subspaces are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid both a $\wedge$ and a $\vee$ as a subposet. We generalize some of the results to brooms and forks, and we also show that the union of the set of subspaces of dimension $k$ and $k+1$, for $k = \left\lfloor n/2 \right\rfloor$ or $k = \left\lceil n/2 \right\rceil-1$, are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid a butterfly (definitions below).
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